In the case of the five strategies listed above, Maynard Smith wanted to know what would happen in a population containing copies of all five programs. Was there one of the five which, if it came to predominate, would retain its numerical superiority against all comers? He concluded that program number 3 is an ESS: when it happens to be very numerous in the population no other program from the list does better than program 3 itself (actually there is a problem over this particular example—Dawkins 1980, p. 7—but I shall ignore it here). When we talk of a program as ‘doing better’ or as being ‘successful’ we are notionally measuring success as capacity to propagate copies of the same program in the next generation: in reality this is likely to mean that a successful program is one which promotes the survival and reproduction of the animal adopting it.
What Maynard Smith, together with Price and Parker (Maynard Smith & Price 1973; Maynard Smith & Parker 1976), has done is to take the mathematical theory of games and work out the crucial respect in which that theory must be modified to suit the Darwinist’s purpose. The concept of the ESS, a strategy that does relatively well against copies of itself, is the result. I have already made two attempts at advocating the importance of the ESS concept and explaining its broad applicability in ethology (Dawkins 1976a, 1980), and I shall not repeat myself unnecessarily here. Here my purpose is to develop the relevance of this way of thinking for the subject of the present book, the debate about the level at which natural selection acts. I shall begin by recounting a specific piece of research that used the ESS concept. All the facts I shall give are from the field observations of Dr Jane Brockmann, reported in detail elsewhere and briefly mentioned in Chapter 3. I shall have to give a brief account of the research itself before I can relate it to the message of this chapter.
Sphex ichneumoneus is a solitary wasp, solitary in the sense that there are no social groups and no sterile workers, although the females do tend to dig their nests in loose aggregations. Each female lays her own eggs, and all labour on behalf of the young is completed before the egg is laid—the wasps are not ‘progressive provisioners’. The female lays one egg in an underground nest which she has previously provisioned with stung and paralysed katydids (long-horned grasshoppers). Then she closes up that nest, leaving the larva to feed on the katydids, while she herself starts work on a new nest. The life of an adult female is limited to about six summer weeks. If one wished to measure it, the success of a female could be approximated as the number of eggs that she successfully lays on adequate provisions during this time.
The thing that especially interested us was that the wasps seemed to have two alternative ways of acquiring a nest. Either a female would dig her own nest in the ground, or she would attempt to take over an existing nest which another wasp had dug. We called these two behaviour patterns digging and entering, respectively. Now, how can two alternative ways of achieving the same end, in this case two alternative ways of acquiring a nest, coexist in one population? Surely one or the other ought to be more successful, and the less successful one should be removed from the population by natural selection? There are two general reasons why this might not happen, which I will express in the jargon of ESS theory: firstly, digging and entering might be two outcomes of one ‘conditional strategy’; secondly, they might be equally successful at some critical frequency maintained by frequency-dependent selection—part of a ‘mixed ESS’ (Maynard Smith 1974, 1979). If the first possibility were correct, all wasps would be programmed with the same conditional rule: ‘If X is true, dig, otherwise enter’; for instance, ‘If you happen to be a small wasp, dig, otherwise use your superior size to take over another wasp’s burrow’. We failed to find any evidence for a conditional program of this or any other kind. Instead, we convinced ourselves that the second possibility, the ‘mixed ESS’, fitted the facts.
There are in theory two kinds of mixed ESS, or rather two extremes with a continuum between. The first extreme is a balanced polymorphism. In this case, if we want to use the initials ‘ESS’, the final S should be thought of as standing for state of the population rather than strategy of the individual. If this possibility obtained, there would be two distinct kinds of wasps, diggers and enterers, who would tend to be equally successful. If they were not equally successful, natural selection would tend to eliminate the less successful one from the population. It is too much to hope that, by sheer coincidence, the net costs and benefits of digging would exactly balance the net costs and benefits of entering. Rather, we are invoking frequency-dependent selection. We postulate a critical equilibrium proportion of diggers, p*, at which the two kinds of wasps are equally successful. Then, if the proportion of diggers in the population falls below the critical frequency, diggers will be favoured by selection, while if it rises above the critical frequency enterers will be favoured. In this way the population would hover around the equilibrium frequency.
It is easy to think of plausible reasons why benefit might be frequency-dependent in this way. Clearly, since new burrows come into existence only when diggers dig them, the fewer diggers there are in the population the stronger will be the competition among enterers for burrows, and the lower the benefit to a typical enterer. Conversely, when diggers are very numerous, available burrows abound and enterers tend to prosper. But, as I said, frequency-dependent polymorphism is only one end of a continuum. We now turn to the other end.
At the other end of the continuum there is no polymorphism among individuals. In the stable state all wasps obey the same program, but that program is itself a mixture. Every wasp is obeying the instruction, ‘Dig with probability p, enter with probability 1 – p’; for instance, ‘Dig on 70 per cent of occasions, enter on 30 per cent of occasions’. If this is regarded as a ‘program’, we could perhaps refer to digging and entering themselves as ‘subroutines’. Every wasp is equipped with both subroutines. She is programmed to choose one or the other on each occasion with a characteristic probability, p.
Although there is no polymorphism of diggers and enterers here, something mathematically equivalent to frequency-dependent selection can go on. Here is how it would work. As before, there is a critical population frequency of digging, p*, at which entering would yield exactly the same ‘pay-off’ as digging. p* is, then, the evolutionarily stable digging probability. If the stable probability is 0.7, programs instructing wasps to follow a different rule, say ‘Dig with probability 0.75’, or ‘Dig with probability 0.65’, would do less well. There is a whole family of ‘mixed strategies’ of the form ‘Dig with probability p, enter with probability 1 – p’, and only one of these is the ESS.
I said that the two extremes were joined by a continuum. I meant that the stable population frequency of digging, p* (70 per cent or whatever it is), could be achieved by any of a large number of combinations of pure and mixed individual strategies. There might be a wide distribution of p values in individual nervous systems in the population, including some pure diggers and pure enterers. But, provided the total frequency of digging in the population is equal to the critical value p*, it would still be true that digging and entering were equally successful, and natural selection would not act to change the relative frequency of the two subroutines in the next generation. The population would be in an evolutionarily stable state. The analogy with Fisher’s (1930a) theory of sex ratio equilibrium will be clear.
Proceeding from the conceivable to the actual, Brockmann’s data show conclusively that these wasps are not in any simple sense polymorphic. Individuals sometimes dug and sometimes entered. We could not even detect any statistical tendency for individuals to specialize in digging or entering. Evidently, if the wasp population is in a mixed evolutionarily stable state, it lies away from the polymorphism end of the continuum. Whether it is at the other extreme—all individuals running the same stochastic program—or whether there is some more complex mixture of pure and mixed individual programs, we do not know. It is one of the central messages of this chapter that, for our research purpose, we did not
need to know. Because we refrained from talking about individual success, but thought instead about subroutine success averaged over all individuals, we were able to develop and test a successful model of the mixed ESS which left open the question of where along the continuum our wasps lay. I shall return to this point after giving some pertinent facts and after outlining the model itself.
When a wasp digs a burrow, she may either stay with it and provision it, or she may abandon it. Reasons for abandoning nests were not always obvious, but they included invasion by ants and other undesirable aliens. A wasp who moves into a burrow that another has dug may find that the original owner is still in residence. In this case she is said to have joined the previous owner, and the two wasps usually work on the same nest for a while, both independently bringing katydids to it. Alternatively, the entering wasp may be fortunate enough to hit upon a nest that has been abandoned by its original owner, in which case she has it to herself. The evidence indicates that entering wasps cannot distinguish a nest that has been abandoned from one that is still occupied by its previous owner. This fact is not so surprising as it may seem, since both wasps spend most of their time out hunting, so two wasps ‘sharing’ the same nest seldom meet. When they do meet they fight, and in any case only one of them succeeds in laying an egg in the nest under dispute.
Whatever had precipitated the abandoning of a nest by the original owner seemed usually to be a temporary inconvenience, and an abandoned nest constituted a desirable resource which was soon used by another wasp. A wasp who enters an abandoned burrow saves herself the costs associated with digging one. On the other hand, she runs the risk that the burrow she enters has not been abandoned. It may still contain the original owner, or it may contain another entering wasp who got there first. In either of these cases the entering wasp lets herself in for a high risk of a costly fight, and a high risk that she will not be the one to lay the egg at the end of a costly period of provisioning the nest.
We developed and tested a mathematical model (Brockmann, Grafen & Dawkins 1979) which distinguished four different ‘outcomes’ or fates that could befall a wasp in any particular nesting episode.
1 She could be forced to abandon the nest, say by an ant raid.
2 She could end up alone, in sole charge of the nest.
3 She could be joined by a second wasp.
4 She could join an already incumbent wasp.
Outcomes 1 to 3 could result from an initial decision to dig a burrow. Outcomes 2 to 4 could result from an initial decision to enter. Brockmann’s data enabled us to measure, in terms of probability of laying an egg per unit time, the relative ‘payoffs’ associated with each of these four outcomes. For instance, in one study population in Exeter, New Hampshire, Outcome 4, ‘joining’, had a payoff score of 0.35 eggs per 100 hours. This score was obtained by averaging over all occasions when wasps ended up in that outcome. To calculate it we simply added up the total number of eggs laid by wasps who, on the occasion concerned, had joined an already incumbent wasp, and divided by the total time spent by wasps on nests that they had joined. The corresponding score for wasps who began alone but were subsequently joined was 1.06 eggs per 100 hours, and that for wasps who remained alone was 1.93 eggs per 100 hours.
If a wasp could control which of the four outcomes she ended up in, she should ‘prefer’ to end up alone since this outcome carried the highest payoff rate, but how might she achieve this? It was a key assumption of our model that the four outcomes did not correspond to decisions that were available to a wasp. A wasp can ‘decide’ to dig or to enter. She cannot decide to be joined or to be alone any more than a man can decide not to get cancer. These are outcomes that depend on circumstances beyond the individual’s control. In this case they depend on what the other wasps in the population do. But just as a man may statistically reduce his chances of getting cancer, by taking the decision to stop smoking, so a wasp’s ‘task’ is to make the only decision open to her—dig or enter—in such a way as to maximize her chances of ending up in a desirable outcome. More strictly, we seek the stable value of p, p*, such that when p* of the decisions in the population are digging decisions, no mutant gene leading to the adoption of some other value of p will be favoured by natural selection.
The probability that an entering decision will lead to some particular outcome, such as the desirable ‘alone’ outcome, depends on the overall frequency of entering decisions in the population. If a large amount of entering is going on in the population, the number of available abandoned burrows goes down, and the chances go up that a wasp that decides to enter will find herself in the undesirable position of joining an incumbent. Our model enables us to take any given value of p, the overall frequency of digging in the population, and predict the probability that an individual that decides to dig, or an individual that decides to enter, will end that episode in each of the four outcomes. The average payoff to a wasp that decides to dig, therefore, can be predicted for any named frequency of digging versus entering in the population as a whole. It is simply the sum, over the four outcomes, of the expected payoff yielded by each outcome, multiplied by the probability that a wasp that digs will end up in that outcome. The equivalent sum can be worked out for a wasp that decides to enter, again for any named frequency of digging versus entering in the population. Finally, making certain plausible additional assumptions which are listed in the original paper, we solve an equation to find the population digging frequency at which the average expected benefit to a wasp that digs is exactly equal to the average expected benefit to a wasp that enters. That is our predicted equilibrium frequency which we can compare with the observed frequency in the wild population. Our expectation is that the real population should either be sitting at the equilibrium frequency or else in the process of evolving towards the equilibrium frequency. The model also predicts the proportion of wasps ending up in each of the four outcomes at equilibrium, and these figures too can be tested against the observed data. The model’s equilibrium is theoretically stable in that it predicts that deviations from equilibrium will be corrected by natural selection.
Brockmann studied two populations of wasps, one in Michigan and one in New Hampshire. The results were different in the two populations. In Michigan the model failed to predict the observed results, and we concluded that it was quite inapplicable to the Michigan population, for unknown reasons as discussed in the original paper (the fact that the Michigan population has now gone extinct is probably fortuitous!). The New Hampshire population, on the other hand, gave a convincing fit to the predictions of the model. The predicted equilibrium frequency of entering was 0.44, and the observed frequency was 0.41. The model also successfully predicted the frequency of each of the four ‘outcomes’ in the New Hampshire population. Perhaps most importantly, the average payoff resulting from digging decisions did not differ significantly from the average payoff resulting from entering decisions.
The point of my telling this story in the present book has now finally arrived. I want to claim that we would have found it difficult to do this research if we had thought in terms of individual success, rather than in terms of strategy (program) success averaged over all individuals. If the mixed ESS had happened to lie at the balanced polymorphism end of the continuum, it would, indeed, have made sense to ask something like the following. Is the success of wasps that dig equal to the success of wasps that enter? We would have classified wasps as diggers or enterers, and compared the total lifetime’s egg-laying success of individuals of the two types, predicting that the two success scores should be equal. But as we have seen these wasps are not polymorphic. Each individual sometimes digs and sometimes enters.
It might be thought that it would have been easy to do something like the following. Classify all individuals into those that entered with a probability less than 0.1, those that entered with a probability between 0.1 and 0.2, those with a probability between 0.2 and 0.3, between 0.3 and 0.4, 0.4 and 0.5, etc. Then compare the lifetime reproductive su
ccesses of wasps in the different classes. But supposing we did this, exactly what would the ESS theory predict? A hasty first thought is that those wasps with a p value close to the equilibrium p* should enjoy a higher success score than wasps with some other value of p: the graph of success against p should peak at an ‘optimum’ at p*. But p* is not really an optimum value, it is an evolutionarily stable value. The theory expects that, when p* is achieved in the population as a whole, digging and entering should be equally successful. At equilibrium, therefore, we expect no correlation between a wasp’s digging probability and her success. If the population deviates from equilibrium in the direction of too much entering, the ‘optimum’ choice rule becomes ‘always dig’ (not ‘dig with probability p*’). If the population deviates from equilibrium in the other direction, the ‘optimum’ policy is ‘always enter’. If the population fluctuates at random around the equilibrium value, analogy with sex ratio theory suggests that in the long run genetic tendencies to adopt exactly the equilibrium value, p*, will be favoured over tendencies to adopt any other consistent value of p (Williams 1979). But in any one year this advantage is not particularly likely to be evident. The sensible expectation of the theory is that there should be no significant difference in success rates among the classes of wasps.
In any case this method of dividing wasps up into classes presupposes that there is some consistent variation among wasps in digging tendency. The theory gives us no particular reason to expect that there should be any such variation. Indeed, the analogy with sex ratio theory just mentioned gives positive grounds for expecting that wasps should not vary in digging probability. In accordance with this, a statistical test on the actual data revealed no evidence of inter-individual variation in digging tendency. Even if there were some individual variation, the method of comparing the success of individuals with different p values would have been a crude and insensitive one for comparing the success rates of digging and entering. This can be seen by an analogy.